# How many prime numbers end with 3

Even after decades of research, mathematicians are still discovering tangible surprises in the realm of prime numbers. This is shown by a recently published work by two number theorists at Stanford University in California. Kannan Soundararajan and his colleague Robert Lemke Oliver discovered a property of prime numbers that suggests that these numbers are not as random as theorists previously suspected.

### Successive prime numbers are reluctant to repeat their final digit

A prime number is an integer that can only be divided by itself and by 1 with no remainder. It is known that prime numbers become rarer with increasing size - simply because there are more possibilities to find a divisor. And prime numbers cannot end in an even number (otherwise they are divisible by 2) or in a 5, otherwise they are divisible by 5. Only 1, 3, 7 and 9 remain as the last digits of a prime number.

These endings generally appear about the same number of times. However, the two Stanford mathematicians have now found that successive prime numbers are reluctant to repeat the last digit. At least for the first trillion prime numbers, the following applies: If a prime number ends with the digit 1, the probability that the next larger prime number also ends with a 1 is only 18 percent. With a frequency of 30 percent, the final digit of the next larger prime number is a 3 or a 7. The frequency of a 9 after a 1 is 22 percent. With a completely random distribution of the end digits, all of these frequencies would have to be 25 percent each, since there are four possible end digits. There is a similar dislike for consecutive, identical final digits for the 3, 7 and 9.

"That is actually surprising," comments mathematics professor Eva Viehmann from the Technical University of Munich. One did not necessarily expect the opposite, says the arithmetic geometry expert, but now it is clear: The last digit of a prime number is not a pure coincidence.

Mathematicians do not yet have a simple explanation for this statistical abnormality. The discovery is eagerly discussed in relevant Internet forums. The discovery has no foreseeable consequences for real-life applications of prime numbers, for example in encryption technology in banking. So far it has been a curiosity - and possibly an indication of other laws hidden in the realm of prime numbers that still need to be explored.

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